1. Field of the Invention
The present invention generally relates to the field of predictive models for ensuring the microbiological safety and extended shelf life of foods and, more particularly, to a Quasi-chemical mathematical modeling method which is utilized as a tool for predictive microbiology evaluation of microorganism population dynamics based on an understanding of chemical reaction pathways that are intrinsic to these organisms in support of the safe design of food product formulations and food processing conditions.
2. Description of the Related Art
To develop effective predictive models that assure the microbiological safety and extended shelf life of foods, it is essential to know how bacterial populations grow and die in response to the factors influencing a food product where nutrients are generally non-limiting. Intrinsic factors of the food include physical properties, such as pH, water activity, salinity, and the presence of anti-microbial constituents. Extrinsic factors generally refer to properties of the external environment, such as storage temperature, relative humidity, ambient pressure, and applied processing conditions, all of which influence microbial survivability.
Early investigators faced with the problem of analyzing bacterial population dynamics turned to equations developed previously developed from theories of treating human and animal population dynamics in their effort to model microorganism growth curves and death curves. These models were characterized by parameters, such as per capita birth rate and sustainable population, which are clearly inappropriate and uninformative for describing the growth and death of unicellular microorganisms. It was not obvious to these early modelers that such observed features of bacterial population growth and decline data could be analyzed and evaluated in a manner that reflects the underlying biochemical and biological bases of these changes.
One early known application of chemical reactions involving molecular species that can be used to model the complex and emergent macroscopic behavior of entire organisms involves the Lotka-Volterra model of predator-prey population dynamics. This model consists of three irreversible chemical reactions. The variables are x (i.e, the population of hares (prey)), and y (i.e., the population of lynxes (predator)). The reactions and their differential rate equations comprise autocatalytic growth of the hare population feeding on a constant supply of grass, autocatalytic growth of the lynx population feeding on hares, and a first-order death of lynxes. Solutions of the model predict oscillations in the two populations of lynx and hare that lag behind each other. In a microbiological application of the Lotka-Volterra model, studies of controlled laboratory populations of mixed infusions of Paramecium aurelia (predator) and Saccharomyces exiguns (prey) exhibit these same properties of oscillation and lag. The use of the Lotka-Volterra model is therefore unsuitable for providing accurate estimates of biologically-relevant kinetics parameters for use in predicting conditions to assure the microbiological safety of foods.
The Quasi-chemical model is a mechanistic-based mathematical model that applies appropriate sequences of chemical reactions or biochemical processes to more accurately and meaningfully represent the molecular mechanisms of bacterial anabolism, catabolism, cellular signaling (i.e., Quorum sensing), and lethality that result in growth-death behavior and offers several technological advantages over anthropomorphic models invoked by early investigators, as mentioned above, or other empirical models currently in use.
In Predictive Microbiology, predictive models provide food technologists and non-mathematical experts with convenient food safety tools to determine the survivability of microorganisms in response to food formulations designed to control growth or in response to process conditions intended to limit or destroy pathogenic bacterial populations wherever they may originate. The data characteristic of bacterial population dynamics are collected, categorized, and referred to in terms of quantitative parameters (e.g., lag time, growth rate, etc.) using mathematical models or equations. Workers skilled in this art are expected to use this information to predict how or whether the population will evolve in time, particularly for conditions not used in the construction of the model.
Once a target microorganism has been identified as a hazard associated with a specific food or food class, whether a vegetative bacterial pathogen, bacterial spore, spoilage organism (yeasts and molds), virus, or fungus, prion, or ascospore, its changes in population size will be influenced by intrinsic and extrinsic factors. If the food product is formulated as a minimally processed foodstuff, then the data for this purpose are collected directly by inoculating the foodstuff under a variety of conditions (various formulations and storage conditions) and sampling the population at timed intervals. Similarly, for canned or packaged food products that are intended for exposure to harsh, lethal processing treatments (e.g., heat, known as thermal processing, canning, or retorting), the extent of microbial inactivation in the inoculated foodstuff is determined by sampling at timed intervals of exposure to the lethal processing treatment until its safety is reliably assured.
For purposes of evaluation and categorization, the preferred dynamical data is obtained by enumerating the bacterial population size at selected time intervals using plate-counting or other such techniques. Bacterial populations are often sufficiently large (on the order of 109 colonies per g or per mL in units of CFU/g or CFU/mL) to afford reproducible measurements and meaningful statistics, although they do not approach the enormous sizes of individual molecules in chemical systems (on the order of Avogadro's number NA=6.023×1023). The method of the present invention of the Quasi-chemical mathematical model yields precisely those parameters that enable the model to be predictive as well as descriptive.
In the nutrient-rich environment of food, bacteria and other organisms tend to exhibit a characteristic pattern called the microbial lifecycle. The bacteria or other organisms can survive or be stimulated to grow after a relatively quiescent initial period with a relatively slow growth rate that generally produces only a small or modest increase in population (called the lag phase). As the bacteria metabolize and reproduce, the growth rate increases dramatically, leading to a relatively sudden increase in population size (called the exponential growth phase). The rate of growth then declines asymptotically to zero, and the population increases in number until the population density reaches an approximately constant maximum stationary value (called the asymptotic or stationary phase). However, this so-called stationary phase is not a true steady-state. Rather the stationary phase is indicative of a microbial population in which the rates of growth and death approximately cancel. As the bacterial population ages further, nutrients deplete and excreted metabolic waste products accumulate in the surroundings, and eventually the population density declines due to natural effects (called the death phase).
Conventional methods involve the fitting of bacterial population curves based on treating the population-time curve as an assemblage of recognizable and separable stages of stasis, increase or decrease. Experimental conditions may actually be arranged to separate growth and inactivation responses by the target microorganism population, and distinctly different models have been applied to evaluate the kinetics of each type of response. Specifically, in known methods, data collected during the first phases of population dynamics (lag through stationary phases) are called growth studies. Typically, a food product is inoculated to a low level with the target pathogen, and the inoculated, nutrient-rich food is packaged and stored in a relevant environment characteristic of the actual handling of the product by distributors and consumers. The growth of that organism is measured as a function of time until the maximum cell population is reached and maintained for some unspecified duration. In some food formulations over the range of conditions tested experimentally, only death of the microorganisms may be observed (i.e, an inactivation plot).
Growth kinetics data is typically evaluated using empirical S-shaped (sigmoidal) expressions. While these sigmoidal equations provide reasonable estimates of growth behavior and food shelf-life, they are flawed, because they subjectively remove data showing a decline after the stationary phase. However, the declining or so-called death data influences the estimated values of the sigmoidal curve-fit parameters determined from the growth curves (e.g., growth rates) that are used to estimate the safe, stable shelf-life of the product. In other types of studies, a food product is inoculated to relatively high levels with a target organism, and then subjected to the extreme conditions of an external intervention intended to kill the target organism (e.g., thermal processing, high pressure, pulsed electric field, irradiation, chemical agents, UV light, etc.) by a food processing technology. For such studies, the bacterial population size shows only a decline (i.e., an inactivation plot) from its original inoculated value. In the known methods, the treatment of this type of data is most commonly performed with a first-order decay equation.
Conventional methods perform a selection from among many different, unrelated empirical equations for modeling growth or death kinetics. Models based on sigmoidal functions are used to fit the growth portion of the curves, and a modified version of a sigmoidal function, or a different function, must be chosen to separately model microbial populations showing death-only kinetics. A choice among empirical mathematical functions such as the Weibull distribution model or the logistic equation must be made to provide a fit to either the growth portion or the death portion of the data. Known methods preferably utilize the actuarial Gompertz equation to fulfill this function for growth modeling. The Gompertz equation, and more sophisticated versions of it, have been widely used and have been made available in the United Kingdom (Food MicroModel), for example, and in the United States by the Department of Agriculture (Pathogen Modeling Program) for growth modeling. Because of the wholesale use of the Gompertz equation for growth curves, an equation leading to a curve of similar form, but one that displays a decline not an increase, was sought for death curves. Such equations include an inverse Gompertz function or a Fermi distribution function for atomic density. The complex pattern of full growth-death curves is conventionally treated in a piecewise or discontinuous manner by combining a growth equation and a death equation to yield a single empirical expression that can be used to fit the entire growth-death curve. Such an approach has been used in instances of food fermentations for the production of wine or table olives, and in bio-reactor technologies where nutrients deplete due to bio-mass conversion. In the case of thermal food processing, models such as first-order chemical or radioactive decay equations are often used.
The foregoing description is represented by those skilled in this field as log-log plots of cell density against time, which depict temporal changes in population density in a closed system and is characterized by the four phases: lag, exponential growth, stationary, and death. This known methodology is flawed, however, because there is no provision of the biochemical steps underlying the bacteriological lifecycle when modeling population dynamics. The known methodologies are further flawed in their application, because the models do not fit the entire growth/death data cycle as a continuum, which it actually is. Instead, the known methodologies resolve the population curve into a series of disconnected phases, some of which are re-combined and fitted by the application of separate models. The mathematical functions used to simulate curves so conceived contain parameters that characterize portions of the population-time curve as discontinuous segments.
It is therefore known to use separate and discontinuous models to represent different regions of the response curve, which, when considered in its entirety, is a continuous dynamical process. The “fit” to each phase or region is therefore only approximate. The skilled person has recognized these shortcomings and has attempted to make improvements, but without changing the basic population dynamics approach. In applications to individual strains of bacterial spores, such as Clostridium botulinum, separate expressions for the different types of response to different conditions have been combined into a single equation. Three so-called phases, i.e., lag, growth, and death, are arbitrarily assigned exponential dependences of population on time, and the exponentials are assembled to form a function that fits the entire curve. A further choice is also known: an equation independent of any physical, chemical, or biological model. For example, the growth-decline curve can be fit with a polynomial regression equation. Artisans having the ordinary level of skill in the field of curve-fitting have recognized that the Gompertz equation could be improved, and have introduced a 4-parameter Gompertz equation in attempts to do so. However, even with this modified Gompertz model, the initial or lag phase is poorly modeled. A linear increase in the logarithm of a population [log(CFU/mL)] is often observed during experiments, but the Gompertz equation is one that continually curves and, therefore, the Gompertz fitting process generally yields an approximate fit that overshoots (i.e., provides values slightly above) and undershoots (i.e., provides values slightly below) the experimental data If Gompertz equations are combined to fit growth and death separately, it then becomes necessary to estimate eight parameters plus a parameter for the time at which the two functions are to be combined.
The advent of novel, non-thermal food processing technologies in recent years, such as high pressure processing, to minimize the over-processing of food compared with thermal processing, which itself can compromise the organoleptic properties of food, induce nutrient degradation and diminish food quality, have re-opened the need to examine alternative non-linear models to ensure the safety of foods for consumers. High pressure processing can inactivate foodborne vegetative bacterial pathogens, resistant bacterial spores, viruses, prions, and ascospores under appropriate processing conditions. The microbial inactivation data does not strictly adhere to linear kinetics, often producing non-linear inactivation kinetics for a range of processing conditions, foodstuffs, and target organisms. Consequently, the linear (i.e., first-order) model commonly used with microbial inactivation by thermal processing is unsuited for complex inactivation kinetics, including those observed with non-thermal processing methods. Despite the widespread acceptability of standard sigmoidal equations in commercial practice for ensuring the safety of foods, a single empirical model can not accommodate the entire range of non-linear inactivation kinetics observed with non-thermal food processing, while providing parameters with meaningful insight into the biochemical mechanism of the observed behavior.
It is thus evident that non-linear models with enough versatility so as to capably reproduce the entire array of complex kinetics patterns observed in microbial growth in minimally processed foods or in the inactivation of microorganisms by high pressure and other non-thermal processing technologies are needed to ensure the safety of foods controlled or processed with these technologies. To make accurate predictions of microbial growth or destruction in such circumstances requires the models to have correct structures with meaningful parameters, so that such models can accurately describe the non-linear kinetics and produce parameter values with good statistical quality to improve the accuracy of the estimates of microbial inactivation.
It is therefore apparent there is a need for an improved model to provide more accurate nonlinear (e.g., growth-decline and death-only) curve fits and more relevant modeling parameters for observed microbiological population dynamics.
The mechanistic-based, enhanced 6-parameter Quasi-chemical model is derived from the mathematics of ordinary differential equation systems, and represents a single model structure, which without necessitating modification, can accommodate continuous growth-death-tailing kinetics of microorganisms in minimally processed foods, without subjectively removing data points to evaluate growth curves. The Quasi-chemical model can accommodate linear death-only kinetics and a range of nonlinear inactivation kinetics patterns in either minimally processed foods or foods processed with heat or non-thermal technologies, such as the emerging technology of high pressure processing. In cases where the observed kinetics can show non-linearities, such as lag times (or shoulders) and/or tailing, the data are not amenable to solution by linear models or by the conventional 4-parameter Quasi-chemical model that can not account for tailing in the observed kinetics.